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Short-time diffusion of charge-stabilized colloidal particles:generic features

Marco Heinen, Peter Holmqvist, Adolfo J. Banchio and Gerhard Nagele

J. Appl. Cryst. (2010). 43, 970–980

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J. Appl. Cryst. (2010). 43, 970–980 Marco Heinen et al. · Short-time diffusion of charge-stabilized particles

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970 doi:10.1107/S002188981002724X J. Appl. Cryst. (2010). 43, 970–980

Journal of

AppliedCrystallography

ISSN 0021-8898

Received 2 April 2010

Accepted 9 July 2010

# 2010 International Union of Crystallography

Printed in Singapore – all rights reserved

Short-time diffusion of charge-stabilized colloidalparticles: generic features

Marco Heinen,a Peter Holmqvist,a Adolfo J. Banchiob and Gerhard Nagelea*

aInstitut fur Festkorperforschung, Forschungszentrum Julich, D-52425 Julich, Germany, andbFaMAF, Universidad Nacional de Cordoba, and IFEG-CONICET, Ciudad Universitaria, 5000

Cordoba, Argentina. Correspondence e-mail: g.naegele@fz-juelich.de

Analytical theory and Stokesian dynamics simulations are used in conjunction

with dynamic light scattering to investigate the role of hydrodynamic

interactions in short-time diffusion in suspensions of charge-stabilized colloidal

particles. The particles are modeled as solvent-impermeable charged spheres,

repelling each other via a screened Coulomb potential. Numerical results for

self-diffusion and sedimentation coefficients, as well as hydrodynamic and short-

time diffusion functions, are compared with experimental data for a wide range

of volume fractions. The theoretical predictions for the generic behavior of

short-time properties obtained from this model are shown to be in full accord

with experimental data. In addition, the effects of microion kinetics, nonzero

particle porosity and residual attractive forces on the form of the hydrodynamic

function are estimated. This serves to rule out possible causes for the strikingly

small hydrodynamic function values determined in certain synchrotron

radiation experiments.

1. Introduction

Dispersions of charged colloidal particles undergoing corre-

lated Brownian motion form a particularly important class of

soft-matter systems ubiquitously encountered in the chemical

industry, food science and biology (Pusey, 1991; Nagele, 1996;

Bowen & Mongruel, 1998; Retailleau et al., 1999; Bowen et al.,

2000; Riese et al., 2000; Koenderink et al., 2003; Gapinski et al.,

2005; Prinsen & Odijk, 2007). The calculation of diffusion

transport properties for these systems is challenging since one

needs to cope, in addition to direct electrosteric and van der

Waals interparticle forces, with the solvent-mediated hydro-

dynamic interactions (HIs). The latter type of interaction is

long-ranged and non-pairwise additive in nondilute systems.

In the present paper, using analytical theory, simulation and

light scattering, we discuss generic features of diffusion in

fluid-like ordered suspensions of charge-stabilized colloidal

spheres, as observed on a short-time colloidal scale. The most

frequently used experimental methods to study the dynamics

of charge-stabilized systems are dynamic light scattering

(DLS) and X-ray photon correlation spectroscopy (XPCS). In

both methods, the dynamic structure factor, Sðq; tÞ, is deter-

mined as a function of scattering wavenumber q and corre-

lation time t. At short times, Sðq; tÞ decays exponentially

according to (Pusey, 1991)

Sðq; tÞ / exp �q2DðqÞt� �: ð1Þ

The short-time diffusion function (Nagele, 1996),

DðqÞ ¼ d0HðqÞ=SðqÞ; ð2Þ

is determined by the ratio of the positive-definite hydro-

dynamic function HðqÞ and the static structure factor SðqÞ. At

zero particle concentration, DðqÞ reduces to the single-particle

diffusion coefficient d0. The hydrodynamic function, HðqÞ,obtained experimentally by the short-time measurement of

DðqÞ in combination with a static scattering experiment

determining SðqÞ, quantifies the influence of the HIs on

colloidal short-time diffusion and sedimentation. Without HIs,

HðqÞ is a constant function equal to one. Any q dependence of

HðqÞ reflects the influence of HIs. In the limit of large wave-

numbers compared to the position, qm, of the principal peak of

SðqÞ, the function HðqÞ becomes equal to the normalized

short-time self-diffusion coefficient, dS=d0, which is smaller

than one when HIs are significant. In the limit of q ! 0, HðqÞreduces to the sedimentation coefficient K ¼ Us=U0. Here, Us

is the mean sedimentation velocity of particles in a uniform

suspension subject to a weak constant force field, and U0 is the

single-sphere sedimentation velocity under the same force

field. For an arbitrary value of q, HðqÞ has the meaning of a

generalized sedimentation (or mobility) coefficient, linearly

relating a spatially periodic force field of wavelength 2�=qacting on the particles to the resulting spatially periodic

particle drift velocities. The most significant value of HðqÞ is its

principal peak height, HðqmÞ, attained at a wavenumber that

almost coincides with the position, qm, of the principal peak of

SðqÞ. The value of HðqmÞ relates to the short-time relaxation of

density fluctuations of wavelength 2�=qm, comparable in size

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to the radius of the dynamic cage of next-neighbor particles

formed around each particle.

A theoretical discussion of HðqmÞ for charged spheres as a

function of the particle volume fraction ’ was given recently

by Gapinski et al. (2010). Therein, it was shown that, because

of the increasing strength of near-field HIs with increasing

concentration, HðqmÞ behaves non-monotonically in ’ at low

salinity, showing an initial increase towards its maximal value

larger than one, followed by a decline for further enlarged ’ to

values that can be smaller than one. This behavior differs from

that of the sedimentation and short-time self-diffusion coef-

ficients, which both decrease monotonically with increasing ’.

Gapinski et al. (2010) provided in addition the universal

limiting freezing line for HðqmÞ, which leads to a useful map of

hydrodynamic function peak values attainable in the fluid

regime.

Our calculations of short-time diffusion properties are

based on the one-component macroion fluid (OMF) model,

which describes the colloidal particles as uniformly charged

spheres with stick hydrodynamic boundary conditions on their

surfaces (Banchio & Nagele, 2008). The spheres interact by a

screened Coulomb potential of Derjaguin–Landau–Verwey–

Overbeek (DLVO) type. The accuracy of two analytical

methods to calculate short-time properties is assessed by

comparison with results from numerically accurate, but

computationally expensive, accelerated Stokesian dynamics

(ASD) simulations (Banchio & Brady, 2003; Gapinski et al.,

2005; Banchio & Nagele, 2008), and with DLS data on charged

silica spheres in a toluene–ethanol mixture.

The first analytical method is the �� scheme introduced by

Beenakker & Mazur (1984). It accounts in an approximate

way for many-body HI contributions, but ignores higher-order

near-field HI contributions and lubrication effects. The ��scheme gives qualitatively good results for HðqÞ throughout

the liquid colloid phase. It can be further improved in its

prediction of HðqÞ when its microstructure-independent self-

part is replaced by an accurate simulation expression

(Gapinski et al., 2005; Banchio & Nagele, 2008).

The second analytical method, referred to as the fully

pairwise additive (full PA) approximation, completely

accounts for HIs on the pairwise level, but it ignores three-

body and higher-order hydrodynamic contributions. This

method is exact for very low values of ’, where HIs are

pairwise additive. Therefore, by comparison with ASD simu-

lation data for the considered short-time property, the full PA

scheme allows us to infer the importance of three-body and

higher-order HI contributions. Unlike the �� scheme, the full

PA method is bound to fail at higher concentrations because

of its complete neglect of many-body HIs. Both analytical

methods require SðqÞ or, alternatively, its real-space analog,

the radial distribution function gðrÞ, as the only input. The

static input is computed using an improved version (Heinen et

al., 2010) of the computationally very efficient penetrating

background-corrected rescaled mean spherical approximation

(PBRMSA) scheme (Snook & Hayter, 1992), discussed

further below. The high accuracy of this largely unknown

analytical scheme is demonstrated by comparison with Monte

Carlo (MC) simulations, and results from the accurate but

numerically far more elaborate Rogers–Young (RY) scheme

(Rogers & Young, 1984).

The theoretical and simulation results for the short-time

properties are compared with our (dynamic) light scattering

data on low-salinity suspensions of charged silica spheres, and

with scattering data on charge-stabilized systems by some

other groups. The results presented in this paper serve to

highlight generic features of short-time diffusion properties of

charged particles, in comparison to those of neutral hard

spheres, and to explore the range of applicability of useful

analytical expressions describing a non-analytical ’ depen-

dence of dS, K and HðqmÞ for small values of ’ and long-range

repulsion (low salinity).

In addition, we discuss qualitatively the changes in HðqÞwhen interaction contributions not included in the OMF

model are operative. We analyze the effect of short-range

attractions caused, for example, by van der Waals forces, the

influence of particle porosity, hydrodynamic screening and the

additional dynamic friction due to the electrokinetic relaxa-

tion of the microionic cloud dragged along by each colloidal

sphere. This discussion serves to scrutinize possible causes for

the strikingly small values for HðqÞ determined in synchrotron

radiation experiments by Robert and collaborators (Robert,

2001, 2007; Hammouda & Mildner, 2007; Robert et al., 2008;

Grubel et al., 2008) for certain low-salinity systems. These

values are considerably smaller than those of neutral hard

spheres. The findings of Robert and co-workers are incom-

patible not only with short-time predictions based on the OMF

model but also with experimental data for all other (low-

salinity) charge-stabilized systems that we are aware of, where

HðqmÞ is found to be always larger than the peak value of hard

spheres at the same ’. We correct an incorrect statement by

Robert (2007) made on the equality of the self-diffusion

coefficient for neutral and charged spheres.

While the present work focuses on short-time dynamics, we

point here to recent DLS experiments showing that a far-

reaching scaling behavior of the dynamic structure factor of

neutral hard spheres (Segre & Pusey, 1996), relating short-

time to long-time dynamics, is approximately valid also for

charged colloids (Holmqvist & Nagele, 2010). Short-time

diffusion in charged colloids is not only an interesting topic in

its own right. Such knowledge is also a prerequisite for a better

understanding of the long-time dynamics.

2. Methods of calculation and experimental details

The presented analytical and computer simulation results for

the short-time diffusion properties and SðqÞ are based on the

OMF model. In this simplifying model, a colloidal sphere and

its cloud of neutralizing microions are described as a

uniformly charged sphere of diameter �, interacting electro-

statically by an effective pair potential, uðrÞ, of DLVO type

(Nagele, 1996):

uðrÞkBT

¼ LBZ2 expð�aÞ

1 þ �a

� �2expð��rÞ

r; r>� ¼ 2a: ð3Þ

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J. Appl. Cryst. (2010). 43, 970–980 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles 971electronic reprint

The screening parameter, �, is given by

�2 ¼ 4�LB njZj þ 2nsð Þ=ð1 � ’Þ; ð4Þ

where n is the colloid number density, ns is the number density

of added 1–1 electrolyte and ’ ¼ ð4�=3Þna3 is the colloid

volume fraction of spheres with radius a. Furthermore, Z is the

effective particle charge in units of the elementary charge e,

and LB ¼ e2=ð"kBTÞ is the Bjerrum length of the suspending

Newtonian solvent of dielectric constant " at temperature T,

where kB is the Boltzmann constant. We assume here that the

counter-ions released from the colloid surfaces are mono-

valent. The factor 1=ð1 � ’Þ is frequently introduced to correct

for the free volume accessible to the microions. Hydro-

dynamically, the colloidal particles are treated as nonperme-

able rigid spheres with stick boundary conditions on their

surfaces. The OMF model captures essential features of

charge-stabilized suspensions, for systems where the short-

range van der Waals forces can be neglected.

As mentioned above, we use two efficient analytical

methods to calculate HðqÞ and its small- and large-q limiting

values K and dS=d0, respectively. The zeroth-order �� method

(Beenakker & Mazur, 1984) invokes a partial re-summation of

the many-body HI contributions. It uses truncated hydro-

dynamic mobility tensors without lubrication corrections. An

extensive comparison (Gapinski et al., 2005, 2007, 2009;

Banchio & Nagele, 2008; Banchio et al., 2006) of the �� scheme

predictions for HðqÞ with ASD simulation results has shown

that it reproduces the HðqÞ of charge-stabilized particles quite

well, to a degree even better than for neutral hard spheres

where lubrication is strong, when in place of the original self-

part the accurate ASD simulation result for dS=d0 is used so

that HðqÞ ¼ dASDS =d0 þH

��d ðqÞ. At smaller ’, one can more

conveniently use the full PA scheme result for dS. This

improvement can be understood from noting that dS is

underestimated by the �� scheme, since the zeroth-order

expression for dS used here does not account for the pair

structure of charged spheres. However, and most importantly,

the wavenumber-dependent distinct part, HdðqÞ, is well

described by the �� scheme (Gapinski et al., 2005; Banchio &

Nagele, 2008). In the following section, we exemplify the

accuracy of the self-part-corrected �� scheme, in comparison

to MC simulations and experimental results for HðqÞ, for the

most interesting case of suspensions of strongly correlated

particles at lower salt content.

The second analytical method, the full PA approximation,

uses tables of numerically precise values for the two-body

mobility tensors provided by Jeffrey & Onishi (1984), Kim &

Mifflin (1985) and Jones & Schmitz (1988). The full PA scheme

is a significant improvement over earlier two-body approx-

imations for HðqÞ, also commonly referred to as PA schemes,

where only a long-distance form of the hydrodynamic pair

mobilities in terms of an ða=rÞ inverse pair distance expansion

truncated after a few terms has been used (Nagele et al., 1994,

1995; Nagele, 1996; Watzlawek & Nagele, 1999; Robert, 2001,

2007). The full PA method becomes exact at very small ’where the HIs are truly pairwise additive. However, it neces-

sarily fails at larger ’ since it disregards three-body and

higher-order hydrodynamic contributions.

The static input SðqÞ, required by both analytical methods, is

obtained using the PBRMSA scheme (Snook & Hayter, 1992).

We have augmented this method by a simple density rescaling

of the screening parameter in equation (4). This rescaling

leads to very accurate structure factors, as shown recently by

an extensive comparison with simulation data, and results

from the RY integral equation scheme (Heinen et al., 2010).

Two examples demonstrating the accuracy of the PBRMSA

SðqÞ are discussed in the following section.

Furthermore, we calculate HðqÞ using the ASD simulation

code of Banchio & Brady (2003), extended to the OMF model

of charged spheres (Gapinski et al., 2005; Banchio et al., 2006;

Banchio & Nagele, 2008). This elaborate simulation method

accounts for many-body HIs and lubrication. The calculated

hydrodynamic function exhibits a pronounced OðN�1=3Þdependence on the number, N, of particles in the basic

simulation box. A finite-size scaling extrapolation procedure,

originally used by Ladd (1990) for neutral spheres, is applied

to extrapolate to the HðqÞ of a macroscopically large system.

The good agreement of the finite-size-corrected ASD HðqÞwith the full PA scheme result at small volume fractions

(’< 10�2), where the latter scheme becomes exact, demon-

strates the validity of Ladd’s finite-size scaling method for

application to charged spheres. Stokesian dynamics simula-

tions of HðqÞ are computationally expensive even when the

accelerated code is used. Therefore, fast approximate schemes

such as the �� method are still in demand, in particular when

various system parameters need to be varied to explore

generic features.

For dilute, low-salinity systems of strongly charged particles,

characterized by qm / ’1=3, very simple expressions with

fractional exponents apply:

K ’ 1 � as’1=3; ð5Þ

dS=d0 ’ 1 � at’4=3; ð6Þ

HðqmÞ ’ 1 þ pm’1=3: ð7Þ

These expressions have been derived using the leading-order

far-field forms of the hydrodynamic mobilities (Nagele, 1996;

Nagele et al., 1994, 1995; Watzlawek & Nagele, 1999). The

coefficients as ’ 1:6–1.8 and at ’ 2:5–2.9 in the expressions

for K and dS=d0 depend to a certain extent on the particle

charge and size. The coefficient pm > 0 related to HðqmÞ also

depends on Z and �a (Gapinski et al., 2010; Banchio & Nagele,

2008). All coefficients are typically larger for more structured

suspensions, signalled by a higher peak value of SðqmÞ. As we

will show, the ’ interval where the expression for dS in

equation (6) applies is broader than the interval for the

collective properties K and HðqmÞ.It will prove useful in what follows to compare the short-

time results for charged colloidal spheres with those of neutral

hard spheres at the same ’. Cichocki et al. (1999, 2002) derived

the following virial expansion results for neutral hard spheres

(HS):

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972 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles J. Appl. Cryst. (2010). 43, 970–980

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KHS ¼ 1 � 6:546 ’þ 21:918 ’2 þOð’3Þ; ð8Þ

dHSS =d0 ¼ 1 � 1:832 ’� 0:219 ’2 þOð’3Þ; ð9Þ

valid to quadratic order in ’. These truncated virial expansion

results fully account for HIs up to the three-body level.

The hydrodynamic function peak height of hard spheres is

given to excellent accuracy by Banchio et al. (1999):

HHSðqmÞ ¼ 1 � 1:35’; ð10Þwith the linear ’ dependence valid up to the freezing transi-

tion concentration.

The short-time experimental data presented in this work

have been obtained using DLS from fluid-ordered and nearly

monodisperse suspensions of negatively charged trimethox-

ysilylpropyl methacrylate (TPM)-coated silica spheres

(Philipse & Vrij, 1988), dispersed in an index-matching 80:20

toluene–ethanol solvent mixture at T = 293 K and LB =

8.64 nm. The particle radius determined by small-angle X-ray

scattering (SAXS) is a = 136 nm and the size polydispersity is

0.06. The salinity of residual 1–1 electrolyte is below 1 mM.

Monovalent counter-ions (hydrated protons) are released into

the solvent from the coated silica surfaces. The studied

charged silica system freezes at ’ ’ 0:16 where SðqmÞ ’ 3:2.

The DLS measurements were made using a light-scattering

setup by the ALV-Laservertriebsgesellschaft (Langen,

Germany) and an ALV-5000 multi-tau digital correlator. We

carefully checked that there is no noticeable multiple scat-

tering.

3. Results

In the following, we present our theoretical and simulation

results for charged colloidal particles based on the OMF

model, and compare them with our light scattering data on

coated charged silica spheres and scattering data by two other

groups. The results for charged colloidal spheres (CS) are

compared in addition with corresponding findings for neutral

hard spheres (HS), to highlight salient differences in the short-

time diffusion properties which are largest when low-salinity

charge-stabilized systems are considered. Therefore, only

charge-stabilized systems of lower salinity are considered. For

the influence of added salt on the short-time diffusion coeffi-

cients and HðqÞ we refer to extensive ASD simulations and

analytical calculations published by Gapinski et al. (2009,

2010), Banchio & Nagele (2008) and Banchio et al. (2008).

Therein, it is shown that the OMF-based diffusion properties

cross over monotonically, with increasing salt concentration,

from their zero-salt values to those of neutral hard spheres.

This expected behavior is reflected also in the static structure

factor.

3.1. Self-diffusion

Fig. 1 includes the prediction by the full PA scheme for the

normalized short-time self-diffusion coefficient, dS=d0, of

charged and neutral hard spheres in comparison with ASD

simulation results and our DLS data for TPM-coated charged

silica spheres. The comparison of the full PA scheme result

with the ASD data allows us to deduce quantitatively the

contribution to dS by the non-pairwise additive part of the HIs

arising from the solvent-mediated interactions of three or

more particles.

The large-q regime related to self-diffusion is usually not

accessible by DLS. Therefore, using an argument put forward

by Pusey (1978), we identify dS approximately as dS ’ Dðq�Þ(crosses in Fig. 1), where q� is the first wavenumber located to

the right of qm for which Sðq�Þ = 1 (see top part of Fig. 3).

Simulations of charged and neutral spheres have shown that

dS is determined in this way to within 5–10% accuracy

(Banchio & Nagele, 2008; Abade et al., 2010a,b). A comment is

in order here on an incorrect proposition by Robert (2007),

who stated that one of the present authors (G. Nagele) has

predicted theoretically that there is no difference in the

concentration dependence of dS for charged and neutral

particles. Quite the contrary, Nagele and co-workers have

shown for charged spheres at low salinity that

dS=d0 ’ 1 � at’4=3 [cf. equation (6)], i.e. dS in these systems

has a fractional ’ dependence qualitatively different from that

of hard spheres (Nagele et al., 1994, 1995; Nagele, 1996). Only

the self-diffusion coefficient of neutral hard spheres can be

described by a regular virial series, with the first two virial

coefficients given in equation (9). The correct second-order

term, �0:219’2, in equation (9) differs even in its sign from the

erroneous result, þ0:88’2, used by Robert (2007).

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J. Appl. Cryst. (2010). 43, 970–980 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles 973

Figure 1Normalized short-time self-diffusion coefficient, dS=d0, of a deionizedsuspension of charged spheres (CS, black) and hard spheres (HS, gray).Crosses: DLS data for TPM-coated charged silica spheres. Black circlesand gray diamonds: ASD results for CS and HS, respectively. Solid blackand gray lines: full PA theory results for CS and HS, using the PBRMSAand Percus–Yevick input for SðqÞ, respectively. Dashed black line:1 � 2:5’4=3. Dashed–dotted gray line: second-order virial result for HS.Inset: dS=d0 as predicted by the full PA scheme (lines), and by ASDsimulation (symbols), with the leading-order far-field HI part for CS(black) and HS (gray), respectively, subtracted off.

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The ’4=3 dependence of dS=d0 � 1 was confirmed both by

the experiments of Overbeck et al. (1999) and by ASD

simulations based on the OMF model (Banchio & Nagele,

2008). The depicted ASD data for a low-salinity system of

charged spheres are overall well described by the fractional ’relation in equation (6) for at ’ 2:5, over an extended range of

volume fractions (black solid line in Fig. 1).

Considering the scatter in the charged silica sphere DLS

data for dS depicted in Fig. 1, their overall ’ dependence is

consistent with the ASD simulation data and the ’4=3 scaling

prediction. The full PA scheme overestimates the strength of

the HIs in nondilute suspensions, because it does not account

for the shielding of the HIs between a pair of particles by other

intervening particles. This is the reason why, at intermediate

and larger concentrations, the ASD and experimental data for

dS are underestimated by the full PA method. Note that the

full PA scheme result for dS=d0 is still well described by the

scaling relation in equation (6), but for a somewhat larger

parameter value of at ’ 2:9. We emphasize here that hydro-

dynamic shielding is a many-body HI effect, which lowers the

strength but not the range of the HIs. It should not be

confused, as in earlier work (Riese et al., 2000), with the

screening of the HIs by spatially fixed particles or boundaries

that absorb momentum from the fluid, therefore causing a

faster than 1=r decay of the flow perturbation created by a

point force (Diamant, 2007).

The hydrodynamic self-mobility related to dS is rather

short-ranged, decaying like 1=r4 for a large separation r of two

spheres. Consequently, the difference between dS and its

infinite dilution value d0 is smaller for charged spheres than

for neutral ones, since electric repulsion disfavors near-contact

configurations. The inset in Fig. 1 shows dS=d0, as obtained by

the full PA scheme and ASD simulations, respectively, but

with the far-field part originating from the leading-order self-

mobility part proportional to 1=r4 subtracted off. According to

the inset, dS is rather insensitive to the near-field two-body

part of the HIs, causing a small increase in dS only. The full PA

scheme reproduces exactly the first-order virial coefficient,

�1.832, of hard spheres given in equation (9). This demon-

strates the high precision of the numerical tables for the

hydrodynamic pair mobilities used in our full PA scheme

calculations. Three-body and higher-order HI contributions

come into play for ’>� 0:08, with an enlarging influence on dS

originating from hydrodynamic shielding.

3.2. Sedimentation

In principle, one needs to distinguish between the short-

time and the long-time sedimentation coefficients, but the

latter is smaller than the first by at most a few percent. The two

coefficients are practically equal in dilute systems where the

two-body HI part dominates.

Fig. 2 includes theoretical, simulation and experimental

results for the (short-time) sedimentation coefficient, K, of

homogeneous systems for charged and neutral spheres. The

key message conveyed by this figure is the qualitative differ-

ence in the ’ dependence of K for charged and neutral

particles. This difference is more pronounced than that for the

self-diffusion coefficient discussed earlier. Charged spheres

sediment more slowly than uncharged ones since near-contact

configurations are disfavored. Thus, stronger laminar friction

takes place between the back-flowing solvent and the solvent

layers dragged along with the settling spheres because of the

stick hydrodynamic boundary condition. The solvent back-

flow is created by a pressure gradient directed towards the

container bottom, which balances the nonzero buoyancy-

corrected total gravitational force on the spheres.

At smaller ’ and low salinity, K is well described by the

nonlinear expression 1 � as’1=3 in equation (5), with a coef-

ficient as ’ 1:6–1.8 depending to some extent on the strength

of the electrostatic pair interactions. The exponent 1=3 arises

from the two-body far-field part of the HIs, which dominates

the near-field part for ’<� 0:08, and the scaling relation,

qm / ’1=3, valid in low-salinity systems for the wavenumber

location of the structure factor peak (Banchio & Nagele,

2008). As a consequence, the ’1=3 concentration dependence

of K is observed both for dilute fluid and crystalline systems of

charged particles.

The experimental results of Rojas-Ochoa (2004) for the

low-salinity sedimentation coefficient of a suspension of

charged polystyrene spheres in an ethanol/water mixture

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974 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles J. Appl. Cryst. (2010). 43, 970–980

Figure 2Sedimentation coefficient, K ¼ Us=U0, of charged spheres at low salinitycompared with that of neutral hard spheres. Experimental DLS/SLS dataare shown for charged polystyrene spheres in an ethanol/water mixture(Rojas-Ochoa, 2004) (open squares) and for our TPM-coated silicaspheres (crosses). Solid (dotted) black lines: �� theory (full PA scheme)results for the low-salinity polystyrene spheres system, obtained using thePBRMSA input for SðqÞ with a fixed charge number Z ¼ 200. Dashed–dotted black line: dS-corrected �� scheme result. Dashed black line:scaling form 1 � 1:71’1=3 according to equation (5). Gray filled circles:hard-sphere simulation results by Ladd (1990). Dashed gray line: second-order virial result for HS given in equation (8). Solid (dotted) gray lines:�� scheme (full PA scheme) results for HS with Percus–Yevick input forSðqÞ.

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(a ¼ 58:7 nm, ns ¼ 1 mM, LB ¼ 1:48 nm), and also our data

for the charged silica spheres system, are in accord with the

OMF-based prediction of a steep (as compared to neutral

spheres) ’1=3-like decay of the sedimentation coefficient. The

experimental values of K in both systems discussed in Fig. 2

have been deduced from small-q DLS and static light scat-

tering (SLS) measurements of DðqÞ and SðqÞ, extrapolated to

q ¼ 0. Therefore, there is an unavoidable scatter in the

extrapolated data, in particular considering that the osmotic

compressibility coefficient Sð0Þ of low-salinity systems is very

small.

In contrast to charged particles, the small-’ dependence of

K is well represented for neutral spheres by a regular virial

expansion. In fact, the second-order virial expression in

equation (8) coincides, for ’<� 0:08, with the simulation data

for KHS reported by Ladd (1990). At larger ’, shielding arising

from the higher-order HI terms comes into play, contributing

to K through the higher-order virial coefficients. Since

shielding is disregarded in the full PA scheme, it notably

overestimates the strength of the HIs for ’>� 0:1. When

applied to concentrations beyond its range of applicability, too

small and eventually even nonphysical negative values for K

are predicted (see the dotted lines in Fig. 2). The reason for

the failure of the full PA scheme at higher ’ is that it

approximates the many-sphere hydrodynamic mobility matrix

in a way that does not guarantee the positive definiteness of

this matrix for all physically allowed particle configurations.

Fig. 2 displays additionally, both for neutral spheres and for

the system parameters of the charged polystyrene spheres

system (Rojas-Ochoa, 2004), the predictions for K by the ��scheme of Beenakker & Mazur (1984) and by the full PA

scheme. The theoretical predictions for the silica system are

not shown in order to not overburden the figure with too many

curves.

For neutral hard spheres, in both analytical methods, the

Percus–Yevick solution for SðqÞ is used as input. To leading

order in ’, we have obtained numerically that KHSPA ¼

1 � 6:546’, in full agreement with the first-order coefficient in

equation (8). For comparison, KHS�� ¼ 1 � 7:339’þOð’2Þ.

Thus, the �� scheme underestimates somewhat the hard-

sphere sedimentation coefficient at small ’. It is less accurate

than the PA scheme at small ’ owing to its incomplete account

of two-body HI contributions, notably its neglect of lubrica-

tion, which plays a role for neutral spheres. Lubrication occurs

in the thin fluid layer between two almost touching spheres in

a relative squeezing or shearing motion. It is more influential

to self-diffusion than to sedimentation, since in the latter case

the (monodisperse) spheres move with equal mean velocities

in the direction of the applied force field. In self-diffusion, on

the other hand, a tagged particle is thermodynamically driven

in a squeezing motion towards particles in front of it. At larger

’, however, the �� scheme prediction for KHS is closer to the

simulation data than the full PA scheme result. We attribute

this to the approximate inclusion of many-body HIs into the

�� scheme which describe hydrodynamic shielding.

For charged spheres and small ’, the full PA scheme result

for K follows precisely the scaling prediction 1 � as’1=3, with

as ¼ 1:71. The (uncorrected) �� scheme captures the overall ’dependence of K at least in a qualitative way, regarding both

the two considered charge-stabilized systems and neutral hard

spheres. Extensive comparisons with lower-q ASD simulation

data of HðqÞ (a representative example is shown in x3.4), show

that the �� scheme has the tendency to somewhat under-

estimate K. The self-part-corrected �� scheme, on the other

hand, overestimates K ’ Hðq � qmÞ. At larger q values,

however, including the principal peak region of HðqÞ, it is in

distinctly better agreement with the simulation data for HðqÞthan the uncorrected version (Banchio & Nagele, 2008).

Indeed, the self-part-corrected �� scheme result in Fig. 2 for

charged polystyrene spheres, with the dS input calculated

using the full PA scheme, lies distinctly above the experi-

mental data for K. For the parameters of the polystyrene

system, a least-squares fit of the calculated sedimentation

coefficients in the range ’ � 0:02 leads to equation (5), with

coefficients as ¼ 1:65, 1.75 and 1.71 for the dS-corrected and

uncorrected �� schemes, and the full PA scheme, respectively.

In accord with the general trends discussed above, the full PA

scheme result for K, which becomes exact at low ’, is

bracketed by the self-part-corrected and -uncorrected ��scheme predictions.

3.3. Diffusion function

We proceed by discussing the short-time diffusion function,

DðqÞ, defined in equation (2), which is measured in short-time

DLS and XPCS experiments. DLS data of its inverse, d0=DðqÞ,are included in the bottom part of Fig. 3, for a low-salinity

system of charged silica spheres at a volume fraction ’ ¼ 0:15

rather close to the freezing transition value. The experimental

data are compared with our ASD simulation result for DðqÞ,the (dS-corrected) �� scheme result where the dS part is taken

from the ASD simulation, and the full PA scheme prediction.

For the two analytic schemes, the PBRMSA input for SðqÞshown in the top part of Fig. 3 was used.

The shape of d0=DðqÞ is similar to that of SðqÞ, since

d0=DðqÞ ¼ SðqÞ=HðqÞ according to equation (2). The analy-

tical PBRMSA scheme predicts a structure factor in excellent

agreement with our MC simulation data and with the SðqÞobtained from the numerically elaborate RY scheme. The

excellent agreement between all SðqÞ depicted in the top part

of Fig. 3, for all displayed wavenumbers, points to the accuracy

of our scattering data. The only adjustable parameter in

calculating SðqÞ has been the effective charge number,

uniquely determined as Z ¼ 190 by the PBRMSA, RY and

MC methods, from matching the experimental SðqmÞ.We recall from equation (2) that Dðq ! 1Þ ¼ dS and

Dðq ! 0Þ ¼ d0K=Sðq ! 0Þ ¼ dc. Here, dc is the short-time

collective diffusion coefficient, which quantifies the initial

decay rate of long-wavelength thermal concentration fluc-

tuations. The short-time dc is only slightly larger than its long-

time counterpart, even when a concentrated system is

considered. The relative osmotic compressibility coefficient,

Sðq ! 0Þ, in the considered low-salinity system is very small,

so that dc is much larger than d0, reflected in Fig. 3 by the low-

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J. Appl. Cryst. (2010). 43, 970–980 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles 975electronic reprint

q values of d0=DðqÞ being close to zero. The function DðqÞattains its minimum at qm. The so-called cage diffusion coef-

ficient, DðqmÞ, characterizes the slow relaxation of density

fluctuations of a wavelength �2�=qm, matching the radius of

the nearest-neighbor shell. With increasing concentration and

pair interactions, the cage stiffens, i.e. it becomes more sharply

structured, as reflected by a smaller DðqmÞ.According to the bottom part of Fig. 3, there is good

agreement between the ASD simulation data for d0=DðqÞ and

the dS-corrected �� scheme prediction with its PBRMSA

input. The ASD simulation peak height is somewhat over-

estimated by the uncorrected �� scheme, which uses too small

a value for the self-diffusion coefficient of charged spheres

(see Fig. 1). For the charged silica particles system considered

here, the experimental peak height of d0=DðqÞ happens to be

somewhat closer to that of the uncorrected �� scheme.

However, the first minimum of d0=DðqÞ to the right of qm is in

better accord with the corrected �� scheme prediction.

To illustrate the failure of the full PA scheme for concen-

trations ’>� 0:1 where many-body HIs are strong, we have

included its prediction in Fig. 3. It deviates from the experi-

mental and simulation data most strongly at q ’ 0 and near

qm, reflecting its overestimation of the HIs at the large volume

fraction ’ ¼ 0:15, by giving too small a value for K and too

large a value for HðqmÞ.

3.4. Hydrodynamic function

In Fig. 4 the experimental findings of Philipse & Vrij (1988)

for the HðqÞ and SðqÞ of a well structured, charge-stabilized

suspension of silica spheres suspended in a 70:30 toluene/

ethanol mixture (" = 10 at T = 298 K), are compared with our

theoretical and simulation predictions based on the OMF

model. The PBRMSA, RY and MC SðqÞ of common charge

number Z = 100, shown in the inset, almost coincide in the

depicted q range, demonstrating the accuracy of the PBRMSA

scheme.

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976 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles J. Appl. Cryst. (2010). 43, 970–980

Figure 4Gray diamonds: DLS and SLS data for HðqÞ and SðqÞ (in inset),respectively, taken from Philipse & Vrij (1988), for a charge-stabilizedsystem at ’ ¼ 0:101, in comparison with corresponding ASD and MCdata predictions (filled black circles). Solid gray and black lines:uncorrected and dS-corrected �� scheme results, respectively. Black solidand dashed gray lines in inset: RY and PBRMSA SðqÞ, respectively, forZ ¼ 100, a ¼ 80 nm, ns ¼ 2 mM and LB ¼ 5:62 nm.

Figure 3Top: static structure factor, SðqÞ, obtained by SLS (crosses) and comparedwith PBRMSA and RY results (lines), and MC simulation data (opencircles) for a common Z ¼ 190. Bottom: short-time inverse diffusionfunction, d0=DðqÞ, for a low-salinity system of charged silica spheres.Crosses: DLS data. Open circles: ASD–MC simulation data. Solid blackand gray lines: uncorrected and dS-corrected �� scheme predictions,respectively. Dotted gray line: full PA method result. The systemparameters are a ¼ 136 nm, ’ = 0.15, ns = 0.7 mM and LB = 8.64 nm.

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There is good overall agreement between the experimental

HðqÞ and the ASD and dS-corrected �� scheme results (with

dS taken from the ASD simulation), on accounting for the

scatter in the experimental data for HðqÞ, which has been

obtained from multiplying the experimental data for DðqÞ by

those for SðqÞ. The dS-corrected �� scheme underestimates to

some extent the ASD HðqmÞ, but except for the precise peak

value the overall shape of HðqÞ is well reproduced. Without dS

correction, the ASD HðqÞ is underestimated at all q, owing to

the fact that the �� scheme predicts too small a value for the

charged-particle dS. Figs. 3 and 4 exemplify that the (self-part-

corrected) �� scheme allows for predicting consistently, and to

almost quantitative accuracy, the short-time generic features

of many charge-stabilized systems including small proteins

and large colloidal spheres.

Low-salinity systems are typically characterized by a peak

value of HðqmÞ larger than one. In a recent study based on the

OMF model, the upper limiting freezing line for HðqmÞ was

derived (Gapinski et al., 2010), from which it follows that

HðqmÞ never exceeds the value of 1.3. However, HðqmÞ in low-

salt systems is not always larger than one. At very low ’, it

increases monotonically according to 1 þ pm’1=3, with a

moderately system-dependent coefficient pm > 0 [see

equation (7)]. At larger ’ where near-field HIs matter, HðqmÞcan pass through a maximum typically occurring at ’ ’ 10�2–

10�1, with an ensuing decline when ’ is further increased.

Provided the system remains fluid at larger ’, such as in

apoferritin protein solutions (Gapinski et al., 2005), HðqmÞ can

reach values smaller than one.

In the OMF model, HðqmÞ is bound from below by the

corresponding peak height of neutral spheres. The latter

decreases linearly in ’ in the whole fluid phase regime [see

equation (10)]. At fixed ’ and with increasing salt content,

HðqmÞ and dS decrease monotonically, with qm shifted to larger

q values, towards the limiting hard-sphere values HHSðqmÞ and

dHSS , respectively. Opposite to this, K increases monotonically

with increasing salinity, for the reasons discussed earlier,

towards its upper hard-sphere limit. In summary, the ordering

relations

HðqmÞ � HHSðqmÞ; dS � dHSS ; K � KHS; ð11Þ

are fulfilled. The equality sign holds for zero particle charge,

Z = 0, and in the infinite salinity limit, �a ! 1. The OMF

model ordering relations in equation (11) are obeyed by a

wide variety of experimentally studied charge-stabilized

systems, including nanosized proteins (Gapinski et al., 2005),

suspensions of compact colloidal particles (Philipse & Vrij,

1988; Phalakornkul et al., 1996; Hartl et al., 1999; Rojas-Ochoa,

2004; Rojas-Ochoa et al., 2003; Gapinski et al., 2009; Banchio et

al., 2006; Holmqvist & Nagele, 2010) and thermosensitive

charged microgel spheres (Braibanti et al., 2010).

In a series of articles, Robert, Grubel and coworkers

(Grubel et al., 2008; Robert, 2007; Riese et al., 2000; Robert et

al., 2008) reported their observation of very small values for

HðqÞ for certain low-salinity suspensions of intermediately

large volume fractions, which they studied by combining

XPCS and SAXS techniques. At all probed wavenumbers,

their HðqÞ are substantially smaller than those of neutral hard

spheres at the same ’. This finding of so-called ultra-small

HðqÞ is incompatible with the OMF model since the first two

ordering relations in equation (11) are violated.

A typical result for an ultra-small HðqÞ with peak height

HðqmÞ ’ 0:47, taken from Robert et al. (2008) for a system of

poly(perfluoropentyl methacrylate) spheres in a water/

glycerol mixture at ’ ¼ 0:18, is shown in Fig. 5, in comparison

with our OMF model-based simulation and theoretical results

for HðqÞ, which predict a peak value for HðqÞ larger than one.

The inset displays the experimentally determined SðqÞ, and the

peak-height-adjusted MC, RY and PBRMSA results obtained

for the common charge value Z ¼ 163 and ns ¼ 16 mM(�a ¼ 1:46). The experimental peak height, SðqmÞ ’ 2:63,

identifies the system as fluid-ordered according to the

Hansen–Verlet freezing rule. There is a visible small-q upturn

in the experimental SðqÞ which is not reproduced by the OMF

structure factors describing purely repulsive particles.

However, this upturn should not be over-interpreted as a sign

of a significantly influential particle attraction since the

experimental SðqÞ of very similar systems of charged poly-

(perfluoropentyl methacrylate) spheres reported by Robert et

al. (2008) do not have such an upturn. Moreover, the peak

position of the experimental SðqÞ in Fig. 5 fulfills the relation

qm ¼ 1:1 2�n1=3 (Banchio & Nagele, 2008), characteristic of

a low-salinity system of electrostatically strongly repelling

particles where van der Waals attraction plays no role. This

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J. Appl. Cryst. (2010). 43, 970–980 Marco Heinen et al. � Short-time diffusion of charge-stabilized particles 977

Figure 5XPCS and SAXS data of HðqÞ and SðqÞ (in inset), taken from Robert etal. (2008), for a low-salinity system of charged poly(perfluoropentylmethacrylate) spheres (CS) with a ¼ 62:5 nm, ns ¼ 16 mM and ’ ¼ 0:18in a water/glycerol mixture at T ¼ 293 K, where " ¼ 62:95 andLB ¼ 0:91 nm (open diamonds). Inset: RY and PBRMSA SðqÞ forZ ¼ 163. Comparison with OMF model-based ASD data, dS-corrected ��and full PA scheme results for HðqÞ, all obtained using Z ¼ 163. Theexperimental HðqÞ is substantially smaller than the ASD HHSðqÞ (graycircles) and the �� scheme result for hard spheres (gray solid line).

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follows also from standard DLVO theory when the parameters

for the electric potential part quoted in the caption of Fig. 5

are used.

Just like in the silica system considered previously, the

dS-corrected �� scheme result for HðqÞ, based on the

PBRMSA SðqÞ input for its distinct part depicted in the inset,

and the precise ASD simulation result for its self-part, is in

overall good agreement with the full ASD simulation result

for HðqÞ. It underestimates the ASD peak height to some

extent, but aside from the precise peak value the agreement

with the ASD HðqÞ is quantitatively good.

For completeness, Fig. 5 shows also the full PA scheme

prediction for HðqÞ. The concentration here is clearly too large

for the full PA scheme to apply, with the consequence that

nonphysical negative values of HðqÞ are predicted for

qa<� 1:2. This shows that for the present system, where

’ ¼ 0:18, HðqÞ is strongly influenced by many-body HIs. Quite

notably, however, for the same particle system, an ultra-small

hydrodynamic function, with HðqmÞ ’ 0:7<HHSðqmÞ ¼ 0:95,

was reported by Robert et al. (2008) at ’ ¼ 0:04, i.e. for a

concentration where two-body HIs dominate.

The experimental peak height in Fig. 5 is considerably

smaller than the peak value, HHSðqmÞ ¼ 0:76, of hard spheres,

the latter calculated according to equation (10). To allow for a

comparison at all probed q values, Fig. 5 includes the ASD and

�� scheme results for the HðqÞ of hard spheres. The hard-

sphere structure factor peak value is SHSðqmÞ ¼ 1:19 at

’ ¼ 0:18.

Grubel, Robert and coworkers originally tried to explain

their observation of strikingly low values for HðqÞ as the result

of HI screening (Riese et al., 2000; Robert, 2001). To support

their assertion, they presented a Brinkman fluid-type calcu-

lation of HðqÞ (Riese et al., 2000), wherein only the leading-

order far-field part of the hydrodynamic pair mobility is

considered, treating the Brinkman screening length as a fitting

parameter. However, in a later experimental–theoretical study

(Banchio et al., 2006), it was pointed out that hydrodynamic

screening does not occur in fluid-ordered, unconfined

suspensions of mobile colloidal particles (see here also

Diamant, 2007). Furthermore, the assumed screening of the

HIs conflicts with the fact that the short-time diffusion and

viscosity properties of many charge-stabilized systems, at

concentrations and interaction parameters similar to those

probed by Robert and co-workers, are well explained by OMF

model-based methods without any necessity to invoke HI

screening. The low-salinity system in Fig. 3, for example, is in

the concentration range where an ultra-slow HðqÞ should be

observable.

More recently, Robert and co-workers retracted their

interpretation of ultra-small HðqÞ as being due to HI screening

(Hammouda & Mildner, 2007). In an alternative attempt to

explain their findings (Robert et al., 2008), they introduced a

correction factor, f ¼ �0=�eff < 1, multiplying the OMF

model-based �� scheme HðqÞ, with the value of f determined

such that the ultra-small experimental HðqÞ is matched

overall. Furthermore, they conjecture that f can be identified

by the ratio of the solvent viscosity and some effective

suspension viscosity �eff, leaving it unspecified, however,

whether �eff should be identified with the high-frequency

viscosity, �1, or with the substantially larger static suspension

viscosity. This ad-hoc modification of the �� scheme lacks a

sound physical basis, since the �� scheme expression for HðqÞdescribes a genuine diffusion property. The values for f

obtained from fitting the ultra-small HðqÞ given by Robert et

al. (2008) are consistent with neither calculated (Banchio &

Nagele, 2008) nor experimental (Bergenholtz et al., 1998)

results for �0=�1. In this context, we note that the generalized

Stokes–Einstein relation, ½DðqmÞ=d0 ð�1=�0Þ ’ 1, relating the

cage diffusion coefficient to the short-time (high-frequency)

viscosity, is valid approximately for neutral spheres only

(Banchio & Nagele, 2008; Abade et al., 2010c,d) but not for

low-salinity suspensions of charge-stabilized particles (Koen-

derink et al., 2003; Banchio & Nagele, 2008).

3.5. Influence of additional interactions

In the following, we analyze the effect on HðqÞ caused by

particle interaction contributions not considered in the OMF

model. On a qualitative level, we discuss the influence of

particle porosity, residual attractive forces and microion

kinetics on the shape of HðqÞ.The effect of particle porosity on the HðqÞ of dense

suspensions of neutral porous spheres has been explored in a

recent simulation study (Abade et al., 2010a,b). A nonzero

solvent permeability has the effect of weakening the HIs, thus

reducing the deviations of HðqÞ at all q from its zero-

concentration limiting value of one. For the same reason, a

suspension of porous particles is less viscous than a suspension

of impermeable ones (Abade et al., 2010c,d). Porosity is less

influential when the particles are charged, since near-contact

configurations are then unlikely. The particles studied by

Robert et al. are only very weakly porous, if at all. Thus,

porosity cannot explain the ultra-small HðqÞs. Conversely,

significant porosity would lead to an overall HðqÞ closer to

one.

An attractive interaction contribution enlarges both Sð0Þand the sedimentation coefficient K (Moncha-Jorda et al.,

2010). The enlargement of the latter is overcompensated by

the former, at least at smaller ’ (van den Broeck et al., 1981).

Thus, in dispersions of moderately charged particles, such as

bovine serum albumin or lysozyme proteins with sufficiently

strong short-range attraction, the collective diffusion coeffi-

cient, dc ¼ d0K=Sð0Þ, can attain values smaller than d0

(Cichocki & Felderhof, 1990; Bowen & Mongruel, 1998;

Bowen et al., 2000; Prinsen & Odijk, 2007).

Opposite to sedimentation, attraction tends to slow self-

diffusion, resulting in smaller values of the short-time and

long-time self-diffusion coefficients (Cichocki & Felderhof,

1990; Seefeldt & Solomon, 2003). Attraction-induced slowing

of self-diffusion is accompanied by an augmentation of the

short-time (high-frequency) and long-time (static) suspension

viscosities (Woutersen et al., 1994). Attraction fosters the

formation of short-lived, transient particle pairs and clusters,

which are better shielded from the solvent backflow so that

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sedimentation is enhanced. In self-diffusion, however, the

mean velocity of a weakly forced particle driven towards its

next-neighbor cage particles becomes smaller with increasing

attraction, owing to the greater tendency of nearby particles to

form a transient cluster. This picture also explains why

attraction-induced slowing of long-time self-diffusion is found

not only in colloidal systems, where HIs are present, but also

in atomic liquids (Bembenek & Szamel, 2000). Sedimentation

is different in the sense that all particles, not just a single

tagged one, are forced to move, on average, in the direction of

the external force. Summarizing, the overall effect of attrac-

tion is to lower the difference Hð1Þ �Hð0Þ and to shift the

peak position qm to larger values.

The data reported by Robert and co-workers for SðqÞ and

HðqÞ give no hint of an appreciable attractive interaction part.

The short-range van der Waals attraction acting between the

particles is masked in low-salinity systems by the strong and

long-ranged electric forces to such an extent that it cannot

influence HðqÞ significantly. Moreover, the experimental SðqÞgiven by Robert and co-workers can be described to good

accuracy by the OMF model-based structure factor. Signifi-

cant attraction would enlarge at low q the gap between the

OMF model and the ultra-small HðqÞ in Fig. 5, instead of

reducing it.

On first sight, the non-instantaneous electrokinetic relaxa-

tion of counter- and co-ions forming overlapping electric

double layers around the charged colloids is a more promising

candidate for causing ultra-small HðqÞ. Indeed, the relaxation

of the microion clouds has a slowing influence on colloid

diffusion, referred to as the electrolyte friction effect. This

effect can lower Hð0Þ (Gapinski et al., 2005) to a smaller

extent, and also the values of the short-time and long-time

self-diffusion coefficients (McPhie & Nagele, 2007). However,

electrolyte friction scales with the ratio, d0=dm, of the free

diffusion coefficient, d0, of the slowly moving colloids relative

to the (mean) free diffusion coefficient, dm, of the small

microions (Retailleau et al., 1999; Gapinski et al., 2005; McPhie

& Nagele, 2007). Because of the huge difference in these two

free diffusion coefficients, it is unlikely that electrokinetics can

explain the strikingly low values for HðqÞ reported by Robert

and collaborators. Whereas the electrokinetic influence on

colloid diffusion is very small for larger colloidal spheres, it

can be significantly strong for small, nanosized macroions such

as proteins.

4. Conclusions

The generic behavior of short-time diffusion properties in

suspensions of charge-stabilized colloidal particles with strong

electrostatic interactions has been studied by simulation and

analytical theory, in conjunction with dynamic light scattering

on charged silica spheres. Our calculations are based on the

OMF model, which interpolates between the limiting cases of

a deionized (low-salinity) system and a system of neutral hard

spheres. Two analytical methods to determine HðqÞ, namely

the full PA scheme and the (self-part-corrected) �� scheme,

have been tested against Stokesian dynamics simulations and

compared with experimental results. The full PA scheme

becomes exact at very low volume fractions but it cannot be

applied to denser systems. The self-part-corrected �� scheme

gives overall good results in the whole liquid phase regime, for

all wavenumbers except those in the low-q regime. The

experimental confirmation of OMF model-based low-’predictions, notably the ’1=3 scaling of K (Rojas-Ochoa, 2004)

and HðqmÞ (Hartl et al., 1999), the ’4=3 scaling of dS=d0

(Overbeck et al., 1999; Holmqvist & Nagele, 2010) and in

addition the drS=d

r0 ¼ 1 � ar’

2 scaling (with ar ’ 1:3) of the

short-time rotational diffusion coefficient drS, normalized by its

zero-concentration limiting value dr0 (Koenderink et al., 2003),

add to the credibility of the OMF model.

A large body of experimental results for DðqÞ and HðqÞ, for

systems of different particle types and sizes, concentrations,

salt contents, and solvents, is well described by the OMF

model, with all the ordering relations in equation (11) satis-

fied. Residual attractive-pair interactions or particle porosity,

and most likely also electrolyte friction, cannot explain the

ultra-small HðqÞ findings of Robert and collaborators. Ultra-

small values of HðqÞ have not been observed in our scattering

experiments, nor in those by our collaborators and various

other groups (Philipse & Vrij, 1988; Phalakornkul et al., 1996;

Rojas-Ochoa et al., 2003; Rojas-Ochoa, 2004).

Information on short-time dynamic properties is indis-

pensable for a better understanding of long-time dynamic

properties such as the static viscosity and the long-time self-

diffusion coefficient. Short-time transport coefficients are

used, for example, as input in mode-coupling and dynamic

density functional theory calculations of long-time properties.

For charge-stabilized systems, we have shown recently

(Holmqvist & Nagele, 2010) that dS and DðqÞ are linked to

their corresponding long-time quantities by a simple,

approximate scaling relation.

MH acknowledges support by the International Helmholtz

Research School of Biophysics and Soft Matter (IHRS

BioSoft). AJB acknowledges financial support from SeCyT-

UNC and CONICET. We thank J. Gapinski and A. Patkowski

for helpful discussions. This work was supported by funds from

the Deutsche Forschungsgemeinschaft (SFB-TR6, project

B2).

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